1 Bulletin of Mathematical Biology, 59, 809{831, 1997
DYNAMICAL CONSEQUENCES OF OPTIMAL HOST-FEEDING ON HOST-PARASITOID POPULATION DYNAMICS VLASTIMIL KR IVAN
Department of Theoretical Biology, Institute of Entomology, Academy of Sciences of the Czech Republic, and Faculty of Biological Sciences USB, Branisovska 31, 370 05 C eske Budejovice, Czech Republic (E-mail:
[email protected])
This study examines the in uence of various host-feeding patterns on host-parasitoid population dynamics. The following types of host-feeding patterns are considered: concurrent and non-destructive, non-concurrent and non-destructive, and non-concurrent and destructive. The host-parasitoid population dynamics is described by the Lotka{Volterra continuous time model. This study shows that when parasitoids behave optimally, i. e., they maximize their tness measured by the instantaneous per capita growth rate, non-destructive type of host-feeding stabilizes host-parasitoid dynamics. Other types of host-feeding, i. e., destructive, concurrent, or non-concurrent do not qualitatively change neutral stability of the Lotka{Volterra model. Moreover, it is shown that the pattern of host-feeding which maximizes parasitoid tness is either non-concurrent and destructive, or concurrent and non-destructive host-feeding depending on the host abundance and parameters of the model. E ects of adaptive choice of host-feeding patterns on host-parasitoid population dynamics are discussed.
Introduction. It is believed that behavioral decisions of parasitoids may have a strong im-
pact on host-parasitoid population dynamics. The e ects of various behavioral decisions of parasitoids on host-parasitoid population dynamics were recently studied: searching for food versus ovipositing (Krivan and Sirot 1997), searching for healthy hosts versus superparasitizing (Sirot and Krivan 1997), ovipositing versus host-feeding (Jervis and Kidd 1986, Yamamura and Yano 1988, Kidd and Jervis 1991a, Kidd and Jervis 1991b, Murdoch et al. 1992, Briggs et al. 1995, Jervis and Kidd 1995). Since host-feeding (i. e., the consumption of host haemolymph by the adult female parasitoid) may have a major impact on host mortality, host-feeding parasitoid species may prove to be good biological control agents (Kidd and Jervis 1989). Host-feeding was also studied in the framework of dynamic state variable models (Collier et al. 1994, Houston et al. 1992). These models predict existence of a threshold egg load below which host-feeding occurs. While these predictions qualitatively agree with experimental results of Collier et al. (1994) who showed that probability of host-feeding declines with the egg load, a similar experiment by Rosenheim and Rosen (1992) does not show such a relationship. In Jervis and Kidd (1986) the following types of host-feeding were considered: concurrent, non-concurrent, destructive and non-destructive. Concurrent feeding means that the female parasitoid uses the same host individual for both feeding and oviposition while non-concurrent means that di erent hosts are used. Destructive host-feeding means that hosts die because of the host-feeding while in the case of non-destructive host-feeding hosts survive. Surveying known facts about 64 species of hymenopteran parasitoids Jervis and Kidd (1986) showed that the most common host-feeding patterns to appear are: non-concurrent and destructive (42 species), and concurrent non-destructive (11 species). Some parasitoids also develop a mixed strategy: concurrent non-destructive/non-concurrent destructive (8 species), or concurrent non-
2 destructive/non-concurrent non-destructive (3 species). The decision of a parasitoid whether to host-feed or to oviposit represents a trade-o : host-feeding increases the number of eggs produced but it may decrease the number of ovipositions if host-feeding is non-concurrent. E ects of various host-feeding patterns on population dynamics were discussed in Kidd and Jervis (1989). The underlying dynamics was described by the discrete-time Nicholson{Bailey model. Based on a series of simulations they concluded that \host-feeding behavior is unlikely to contribute to population regulation" since their models exhibit unstable oscillations of increasing amplitude as in the original Nicholson{Bailey model. This result supports the idea of Flanders (1953) that host-feeding results in a low persistence of the parasitoid population when hosts are scare and produces population oscillations of high amplitude. A similar result was also obtained by Briggs et al. (1995) who found that host-feeding per se does not a ect stability of a host-parasitoid system described by a continuous-time Lotka{Volterra model. However, they showed that if the parasitoid mortality rate is a decreasing function of the egg load, then the model may stabilize. Moreover, the input of nutrients from a non-host source has a stabilizing e ect while the use of host material for maintenance is destabilizing. In Yamamura and Yano (1988) the Lotka{Volterra continuous type model with time delay was considered. In this model host-feeding increased the speed of convergence of trajectories to an equilibrium. In the present paper we want to compare the e ects of various host-feeding patterns on host-parasitoid population dynamics described by the Lotka{Volterra continuous-time model. We assume that parasitoids behave in order to maximize their tness as measured by an instantaneous per capita growth rate. We nd the optimal host-feeding strategy for each host-feeding pattern and we study the e ects of optimal behavioral decisions of parasitoids on host-parasitoid population dynamics. Our model predicts that while the destructive type of host-feeding has little in uence on host-parasitoid dynamics, non-destructive host feeding strongly a ects the population dynamics and may lead to a stable equilibrium. We also discuss the e ect of adaptive selection of host-feeding patterns on host-parasitoid population dynamics.
Fitness and optimal host-feeding. In this part we describe parasitoid tness with respect to
various host-feeding patterns. The host abundance is denoted by x and the parasitoid abundance by y. By we denote the search rate of a parasitoid, u denotes the probability that upon an encounter with a host a parasitoid will host-feed, e is the net energy (or protein, depending on the currency used) gained from feeding on one host, o is the energy needed for producing one egg, and s is the maintenance energy needed in unit of time. If N is the number of hosts required to be fed on to satisfy energy demands (i. e., maintenance and egg production) of a parasitoid in unit of time, following Jervis and Kidd (1986) we have N = s + oE ;
e
where E is the number of eggs produced in unit of time. Assuming that during unit time a parasitoid will encounter x hosts and it will host-feed on them with probability u, the total number of hosts which will be host-fed is N = ux: We note that in the above derivation we neglect the time which a parasitoid devotes to handling one host. If ux is smaller than the maintenance energy s, no eggs are produced. This gives the maximum number of eggs E produced in unit of time: uxe ? s E = max o ;0 : First, we consider the case of non-concurrent host-feeding. In this case a parasitoid upon an encounter with a host either host-feeds or oviposits provided it has some eggs. We assume that
3 parasitoids that do not have eggs always feed on hosts upon an encounter. Neglecting the time delay between egg production and oviposition, the maximum number of ovipositions in unit time is limited by the number of encounters with hosts and the number of eggs produced. If hostfeeding and oviposition are mutually exclusive (which is the case of non-concurrent host-feeding) only hosts that are not host-fed may be oviposited. Since the number of hosts encountered in unit time by a single parasitoid which are not host-fed is (1 ? u)x, the maximum number of ovipostions in unit time is constrained by the number of eggs produced and the number of hosts encountered which can be oviposited, i. e., uxe ? s min max o ; 0 ; (1 ? u)x : Now we consider concurrent host-feeding. In this case parasitoids may both host-feed and oviposit in the same host. Since all encountered hosts may be oviposited, the maximum number of ovipositions in unit time is: uxe ? s min max o ; 0 ; x : We de ne tness as the number of o springs produced by an average parasitoid in unit time. This means that only those ovipositions which lead to the emergence of a new parasitoid add to parasitoid's tness. In what follows we consider the following host-feeding patterns: nonconcurrent and destructive (ND), non-concurrent and non-destructive (NN), and concurrent and non-destructive (CN). We do not consider concurrent and destructive host-feeding since in this case oviposited hosts are also host-fed which necessarily reduces tness of this rarely observed host-feeding pattern (Jervis and Kidd 1986). We make the following assumptions. First, we assume that the energy gain from destructive host-feeding is higher than from non-destructive feeding, i.e.,
eD > e N ; where eD denotes the net energy gained from feeding on a host when host-feeding is destructive and similarly for eN . Second, we assume that non-destructive feeding has no ill e ects on the parasitoid's developing progeny. For a xed host abundance x and host-feeding strategy u, the number of ovipositions in unit time, i. e., the parasitoids tness, is given by (the subindex refers to the type of host-feeding pattern considered): uxe ? s D ; 0 ; (1 ? u)x ; fND (x; u) = min max uxeo ? s N ; 0 ; (1 ? u)x ; fNN (x; u) = min max uxeo ? s N fCN (x; u) = min max o ; 0 ; x : We assume that parasitoids behave as to maximize their tness with respect to host-feeding, and for each host abundance we de ne the maximal possible parasitoid tness
Fi(x) = 0max f (x; u); i = ND, NN, CN: u1 i Thus, Fi (x) is the maximal possible tness of a parasitoid measured through the number of o spring produced in unit of time. The values of u which maximize fi(x; u) de ne the optimal strategy of parasitoids with respect to host-feeding.
4 We set
u1D = xes ; D ox u2N = xe+ s ; N s 1 xD = e ; D s 1 xN = e ; N 2 xD = (e s? o) ; D 2 xN = (e s? o) : N
Since tness functions fi ; i = ND; NN; CN are piecewise linear in u, it is easy to determine the optimal strategy for each type of host-feeding. For non-concurrent and destructive host-feeding the optimal strategy is (see Appendix A and Fig. 1): If x < x1D then FND (x) is identically zero since the energy obtained through host-feeding does not meet the amount of energy necessary for the maintenance and no eggs are produced. Consequently, the optimal strategy uopt cannot be uniquely determined, thus 0 uopt 1. In this case there must be another source of energy (like pollen, honeydew etc.) which allows parasitoids to survive. Whether a parasitoid will host-feed or not may depend on the distribution of the alternative food type. If a parasitoid encounters hosts when searching for the alternative food type it is likely that it will host-feed on them (i. e., uopt = 1). If the alternative food is located at a place where are no hosts, parasitoids will have no possibility to host-feed. xeD ? s +s If x x1D then uopt = xox (eD + o) and FND (x) = eD + o : In this case the egg production rate equals the encounter rate with host which are not host fed. The strategy for non-concurrent and non-destructive host-feeding is obtained from the optimal strategy for non-concurrent and destructive host-feeding by replacing index D by N . The optimal strategy for concurrent and non-destructive host feeding depends on the relation between o and eN . First we assume eN > o which means that feeding on one host is sucient to produce more than one egg. Under this assumption the optimal strategy is, see Appendix A: If x < x1N then the optimal strategy is not uniquely determined, 0 uopt 1 and FCN (x) = 0 which means that no eggs are produced because hosts are scare.
If x1N x x2N then uopt = 1 and FCN (x) = xeNo ? s : In this case the number of hosts is too small to produce enough eggs to oviposit each encountered host due to the loss of
energy which goes to the maintenance. A convenient strategy for parasitoids would be to obtain additional energy to cover the maintenance cost from non-host food sources which would allow for higher egg production and for a higher parasitoid growth rate. If x > x2N then the optimal strategy is not uniquely determined, u2N uopt 1 and FCN (x) = x: The number of ovipositions is limited by the number of hosts available for oviposition rather than egg production. In this case the egg production rate exceeds the oviposition rate if uopt > u2N which would lead to the growth of the egg load in ovaries. It is therefore reasonable to assume that if parasitoids have small (or zero) egg load then they may increase host-feeding (i. e., uopt may be higher than u2N ) which will increase the average
5 egg load. If the average egg load reaches the maximum storage capacity of ovaries then uopt = u2N since in this case the number of eggs produced will balance the number of eggs deposited keeping the egg load constant. Second we assume that feeding on one host is not sucient to produce one egg, i. e., eN o which gives the following optimal strategy (see Appendix A): If x < x1N then 0 uopt 1 and FCN (x) = 0 due to the host scarcity.
If x x1N then uopt = 1 and FCN (x) = xeNo ? s : In this case the oviposition rate is always limited by the egg production.
E ects of host-feeding patterns on population dynamics. In this part we study the e ects
of host-feeding patterns on host-parasitoid population dynamics. We assume that parasitoids host-feed optimally, i. e., they maximize their tness, and for a given parasitoid species the host-feeding pattern is xed. The underlying population dynamics is of the Lotka{Volterra continuous time. Non-concurrent and destructive host-feeding. This host-feeding pattern is the most common type observed among hymenopteran parasitoids, (Jervis and Kidd 1986). All encounters of a parasitoid with hosts are lethal for hosts either due to oviposition or host-feeding, i. e., the functional response is x. Assuming that a parasitoid lays one egg per a host which gives to rise FND (x) new parasitoids in unit of time, we get the following continuous-time dynamics:
x0 = x(a ? y); (1) y0 = FND (x)y ? my: Here a is the intrinsic per capita growth rate of hosts and m is the intrinsic per capita mortality rate of parasitoids. Thus we do not consider any density dependence in the growth of the host population. The dynamics of (1) is described for x > x1D by:
and for x x1D by
x0 = x(a ? y); xe ? s D 0 y =y e +o ?m ; D
(2)
x0 = x(a ? y); y0 = ?my:
(3)
It is proved in Appendix B that trajectories of (1) are closed curves centered at the neutrally stable equilibrium of (2) m(e + o) + s a D 2 ; ; E = e D
see Fig. 2. Qualitatively, trajectories look the same as for the classical Lotka{Volterra model. If D ?s a trajectory moves in the part of the space where x > x1D , the egg production rate is xe eD + o + s : The ratio of ovipositions/feeding attacks and the probability of host-feeding is xox (eD + o) 1 ? uopt = xeD ? s for x x1 D uopt ox + s
6 is declining with declining host abundance which corresponds to observations (Kidd and Jervis 1991b). If x < x1D no eggs are produced because all energy available is used for maintenance and parasitoid population is declining which reduces pressure on host population that can recover. Concurrent and non-destructive host feeding. In this case only those hosts which are parasitized will die since host-feeding is non-destructive. Assuming that each parasitoid lays one egg per a host, the functional response is given by the number of ovipositions FCN (x) which is also the numerical response. This leads to the following model: x0 = ax ? FCN (x)y; (4) y0 = FCN (x)y ? my: First we assume that eN > o, i. e., feeding on one host leads to the production of more than one egg. The dynamics of (4) driven by the optimal strategy is described for x > x2N by:
for x1N x x2N by
and for x < x1N by
x0 = x(a ? y); y0 = y(x ? m);
(5)
x0 = ax ? xeNo ? s y; y0 = xeNo ? s y ? my;
(6)
x0 = ax; y0 = ?my:
(7)
If the host abundance is below x1N then the host population is growing since parasitoids do not have enough energy to produce eggs. Thus, all trajectories for which the initial host abundance is below x1N enter in a nite time the region x > x1N and stay there forever. Equation (5) has equilibrium a E5 = m ; and (6) has equilibrium
E =
mo + s a(mo + s)
eN ; eN m : If (8) m < e s? o N then E 5 , E 6 are to the left of the line x = x2N and E 6 is globally asymptotically stable for (4), 6
see Appendix C and Fig. 3A. The mechanism which drives trajectories to the stable equilibrium is the following. If a trajectory of (4) reaches the line x = x2N below the point y = a= from the left then it will cross this line and in the region x > x2N it will move along a Lotka{Volterra cycle centered at E 5 . Since equilibrium E 5 is to the left of the line x = x2N , the trajectory will necessarily return to the region x < x2N where it is attracted by E 6 . In this way trajectories will converge to E 6 and host-feeding has a stabilizing e ect. At the equilibrium the egg deposition rate equals m and parasitoids will play pure strategy (u = 1), i. e., they will host-feed upon each encounter with a host. If m > s=(eN ? o) then E 5 , E 6 are to the right of the line x = x2N and it is proved in Appendix C that qualitative behavior of trajectories depends on initial conditions. If the initial
7 condition belongs to the set bounded by the largest Lotka{Volterra cycle of (5) which is to the right of the line x = x2N then the corresponding trajectory will follow a Lotka{Volterra cycle of (5) (see the small amplitude cycle in Fig. 3B). If initial conditions are outside this set, the corresponding trajectory will converge to the largest Lotka{Volterra cycle which is to the right of the line x = x2N , see Fig. 3B. In this case host-feeding has partially stabilizing e ect on population dynamics in the sense that there exists a bound on population uctuations. The amplitude of maximal uctuations is proportional to the distance of E 5 from the line x = x2N . This distance is given by m(eN ? o) ? s : (eN ? o) For example, uctuations in the host-parasitoid population dynamics increase with increasing instantaneous predator mortality rate m. If we assume eN o, the host-parasitoid dynamics is described by (6) for x > x1N , by (7) for x x1N , and all trajectories converge to E 6 which is globally stable, see Appendix C. Non-concurrent and non-destructive host-feeding. In this case only those hosts which are oviposited will die, which leads to the following host-parasitoid dynamics
x0 = ax ? FNN (x)y; y0 = FNN (x)y ? my:
(9)
The dynamics of (9) driven by the optimal strategy is described for x > x1N by N ? s y; x0 = ax ? xe e +o xe N? s N 0 y =y e +o ?m ; N
(10)
and for x x1N by (7). System (9) has a globally asymptotically stable equilibrium which is the equilibrium of (10) (see Appendix C) m(e + o) + s m(e + o) + s N 10 E = ;a N :
eN
eN m
Trajectories qualitatively look the same as those in Fig. 3A. Host-feeding stabilizes in this case the Lotka{Volterra dynamics. At equilibrium partial preferences for host-feeding appear since uopt = m(emo++o)s + s : N EFFECTS OF ADAPTIVE HOST-FEEDING ON POPULATION DYNAMICS In this section we will assume that the host-feeding patterns may be adaptive. This means that the host-feeding pattern which gives the highest tness should be selected for. We nd which type of the host-feeding pattern gives the maximum tness. We note that
fND (x; u) fNN (x; u); fCN (x; u) fNN (x; u) for any host abundance and any host-feeding strategy (we recall that we assume eD > eN ). This implies FND (x) FNN (x); FCN (x) FNN (x);
8 see Fig. 1. Assume
o < e eN?eDe ; D
and denote
N
(11)
x = (e e ?seoD(e ? e )) : N D D N
We note that for o < eN inequality (11) is satis ed. We get that FCN (x) > FND (x) if x > x , and FCN (x) FND (x) otherwise, see Fig. 1A. Thus, if hosts are abundant, concurrent and non-destructive host-feeding should be selected for while if the host abundance is below x , nonconcurrent and destructive host-feeding should prevail. The optimal parasitoids behavior leads to switching between the concurrent and non-destructive, and non-concurrent and destructive host-feeding with respect to changing host abundance. If o > eN eD =(eD ? eN ) then FND (x) > FCN (x) for all parasitoid densities above x1D , see Fig. 1B, i.e., the non-concurrent and destructive strategy is always the best host-feeding pattern. Thus, the two best possible host-feeding patterns are non-concurrent and destructive, and concurrent and non-destructive host-feeding which qualitatively agrees with observations (Jervis and Kidd 1986). Now we will consider the host-parasitoid dynamics in the case in which switching between concurrent and non-destructive, and non-concurrent and destructive host feeding occurs, i.e., (11) holds. Let vi (here i = ND,CN and vCN + vND = 1) be the probability that host-feeding pattern i is chosen by a parasitoid. The corresponding population dynamics is described by
x0 = ax ? vND xy ? vCN FCN (x)y; y = ?my + vND FND (x)y + vCN FCN (x)y; 0
(12)
and the average tness per parasitoid is
F (x) = vND FND (x) + vCN FCN (x): The optimal strategy is the vector (vND (x); vCN (x)) which maximizes F (x):
If x > x then the best host-feeding pattern is concurrent and non-destructive host-feeding,
i. e., vCN (x) = 1; vND (x) = 0: If x = x then the optimal strategy is not uniquely given and vND (x) 2 [0; 1]; vCN (x) = 1 ? vND : If x1D < x < x then the best host-feeding pattern is non-concurrent and destructive hostfeeding, i. e., vND (x) = 1; vCN (x) = 0: If x < x1D then the optimal strategy is not uniquely de ned, vND (x) + vCD (x) = 1. The reason why for host densities below x1D the host-feeding strategy is not uniquely determined is due to the fact that for all possible host-feeding scenarios the energy obtained from host-feeding is lower than is the maintenance energy and, consequently, no eggs are produced. We will assume, that for low host abundances destructive host-feeding will prevail since it provides more energy for parasitoids to satisfy their energy requirements than non-destructive host-feeding. Thus, for a host abundance below x1D , the dynamics is described by (3). In what follows we will assume that eN eD =(eD ? eN ) > eN > o, and we note that
x1D < x1N < x < x2N :
9 Population dynamics which corresponds to the optimal choice of host-feeding pattern is described by (2) for x1D x x , by (6) for x < x < x2N , and by (5) for x2N < x. If
eN ) m < e e s(?eDo? (eD ? eN ) D N
(13)
we show, that adaptive host-feeding may destabilize the host-parasitoid dynamics. Let us consider a segment of the line x = x which is between the points
y1 = a ; y2 = (e ae?D e ) : D N
Trajectories starting from a point which is close to this segment move away from this segment, see Appendix D. For this reason trajectories which start from the segment are not uniquely de ned since they may move either to the left or to the right. Consider initial densities of host and parasitoids which are close to E 2 . Since due to (13) E 2 is to the left of the line x = x , the corresponding dynamics is described by (2) and follows a Lotka{Volterra cycle, see Fig. 4A. Thus E 2 is neutrally stable. If the initial densities are such that the corresponding trajectory intersects with the line x = x then it cannot return to the region where x < x below the point y2 , see Appendix D. This results in destabilization of the dynamics and trajectories spiral away from the equilibria E 2 , see Appendix E and Fig. 4A. We note, that in this case parasitoid will switch the host-feeding pattern between non-concurrent and destructive, and concurrent and non-destructive periodically with respect to changing host abundance. If opposite inequality to (13) holds and m < s=(eN ? o) then E 2 ; E 6 are to the right of the discontinuity line x = x , and E 6 is to the left of the line x = x2N . In this case E 6 is locally stable but numerical simulations as those given in Fig. 4B suggest that E 6 is not globally stable. If m > s=(eN ? o) then all trajectories do converge to the global attractor given by the largest Lotka{Volterra cycle of (5) which is to the right of the line x = x2N , see Appendix E. The case for which eN eD =(eN ? eD ) > o > eN leads to similar results, see Appendix E. In this case we note that eN < 0 and the dynamics (12) is described by (2) for x1D x x , and by (6) for x < x. If (13) holds then the dynamics is destabilized in a similar way as in Fig. 4A. DISCUSSION In this paper we studied the e ects of various host-feeding patterns on host-parasitoid population dynamics described by the Lotka{Volterra model assuming that parasitoids maximize their tness. We showed that while the destructive type of host-feeding does not qualitatively in uence host-parasitoid population dynamics, non-destructive host-feeding has strong e ect on population dynamics since it leads either to a stable equilibrium, or it reduces the amplitude of maximal uctuations in population densities. Since we did not include in our models any density dependence or other mechanisms which could itself lead to a stable equilibrium, our model predicts that non-destructive host-feeding per se may stabilize neutrally stable Lotka{ Volterra dynamics. Briggs et al. (1995) considered the e ect of the egg load on host-feeding and host-parasitoid dynamics for non-concurrent and destructive host feeding. They assumed, that per capita parasitoid mortality rate is an exponentially decreasing function of egg load. Without the use of eggs for maintenance, host feeding does not qualitatively change behavior of the underlying Lotka{Volterra continuous time model which agrees with prediction from our model. However, when maintenance was included in the Briggs et al. (1995) model, the result switched from no result on stability to a destabilizing e ect. Instead, in our model we assumed that
10 per capita parasitoid mortality rate is constant, because maintenance requirements can be met by other non-host resources like pollen, honeydew, nectar etc. and the stability (for destructive type of host-feeding) is achieved through non-zero maintenance cost s. We note that if parasitoids may obtain energy for maintenance from other sources, or if there are no maintenance requirements, i. e., when s = 0, the stabilizing e ect of non-destructive host feeding is lost and population densities would uctuate around an equilibrium following the Lotka{Volterra model. In Yamamura and Yano (1988) e ects of host-feeding on population dynamics were studied in the framework of the Lotka{Volterra model with intraspeci c competition in host population. They showed that for a positive intraspeci c competition trajectories of the model converge to equilibrium while for zero intraspeci c competition ecological equilibrium is neutrally stable. These results are qualitatively the same as in the case of the classical Lotka{Volterra equation without host-feeding. In the ecological literature the decision whether to host-feed or to oviposit is often interpreted as a trade-o between future and current reproduction (Collier et al. 1994, Heimpel and Rosenheim 1995). On the level of individuals this trade-o leads to dynamical programming which predicts the best feeding behavior of a parasitoid. The general prediction of these models is that there is a threshold in the egg load; if the egg load is below the threshold, host-feeding appears (Collier et al. 1994). However, these models do not consider population dynamics. The main di erence between the dynamic approach and our model is in the de nition of tness. The dynamic approach de nes tness as the overall number of eggs deposited over the lifespan of a parasitoid, while in our interpretation tness is measured through the instantaneous rate of egg laying. Our choice of tness corresponds to the the classical rate-maximizing theory (Stephens and Krebs 1986, Abrams 1983) where tness is de ned as the instantaneous rate of increase of the number of genotype copies, which is commonly used to measure the advantages of life history traits (Stearns 1992). Although this approach is more coarse when compared with the dynamic programming approach, it allows the consideration of population dynamics. We found that the host-feeding pattern which maximizes parasitoid tness is either nonconcurrent and destructive or concurrent and non-destructive host-feeding depending on the host abundance and parameters of the model. If the amount of energy required to mature one egg is high then non-concurrent and destructive host-feeding is the host-feeding pattern which maximizes parasitoid tness. If the amount of energy needed to mature one egg is low then for low host abundances the non-concurrent and destructive host-feeding pattern maximizes parasitoid tness while when hosts are abundant concurrent and non-destructive host-feeding is the best choice. These predictions correspond to the survey by Jervis and Kidd (1986) who found that the most common patterns of host-feeding are, in descending order, non-concurrent and destructive, concurrent and non-destructive, and concurrent and non-destructive/non-concurrent and destructive. In view of these observations, our models suggest that either host abundance for most host-parasitoid systems is low, or the amount of energy needed to mature one egg is high since these are the conditions predicting occurrence of the non-concurrent and destructive host-feeding pattern. We showed, that when the host-feeding pattern is chosen in an adaptive way, this may destabilize host-parasitoid population dynamics. Traditionally, host-parasitoid interactions were modeled by the discrete-time NicholsonBailey model, but continuous-time models based on the Lotka{Volterra like dynamics were also used (Briggs et al. 1995, Yamamura and Yano 1988, Murdoch et al. 1992, Murdoch and Stewart-Oaten 1989). Inclusion of optimal host-feeding behavior of parasitoids lead in this article to non-smooth di erential equations and di erential inclusions. A theory for such types of problems has been recently developed (Filippov 1988, Aubin and Cellina 1984, Aubin 1991, Deimling 1992) and it seems that this methodology may prove to be useful when modelling the e ects of behavioral trade-o s on population dynamics (Krivan 1996, Krivan 1997, Sirot and Krivan 1997, Krivan and Sirot 1997). Although we used the simplest possible type of population dynam-
11 ics in this article, which allowed for a deeper mathematical analysis, the present methodology may be also used for more realistic models, including for example Holing second type functional response function, delays etc. ACKNOWLEDGMENTS The author thanks C. Briggs for his comments and useful suggestions on an earlier version of the manuscript, and E. Sirot for sending some papers on host-feeding phenomena. The stay of the author at Faculty of Biological Sciences was supported by MSMT C R Grant No. VS96086.
12 APPENDIX A: Computation of the optimal strategies First we consider the non-concurrent and destructive host-feeding pattern. For x < x1D and for every u 2 [0; 1] we have (uxeD ? s)=o < 0; thus fND (x; u) = 0 and the corresponding optimal strategy u cannot be uniquely determined. If x > x1D then (uxeD ? s)=o 0 for u 2 [u1D ; 1]. Since (uxeD ? s)=o is an increasing function of u, while (1 ? u)x is decreasing , fND (x; u) is maximized at the point uopt where these two lines intersect, i. e., +s : uopt = xox (eD + o) For the concurrent and non-destructive host-feeding pattern (uxeN ? s)=o < 0 for every u 2 [0; 1] if x < x1N . Therefore fCN (x; u) = 0 if x < x1N and optimal strategy uopt cannot be uniquely de ned. Let us assume that eN > o. If x < x2N then (uxeN ? s)=o < x for every u 2 [0; 1]. Thus uopt = 1 and FCN (x) = (xeN ? s)=o. If x2N < x 8 < uxeN ? s if u < u2 N (14) fCN (x; u) = : o 2 x if u u : N
Thus for x2N < x, fCN is maximized at any uopt which satis es u2N uopt 1 and FCN (x) = x: If eN < o then (uxeN ? s)=o < x for every u 2 [0; 1], i. e., fCN (x; u) = maxf(uxeN ? s)=o; 0g. In this case fCN is maximized at uopt = 1 when x > x1N . If x x1N then fCN (x; u) = 0 for every u 2 [0; 1]. APPENDIX B: Qualitative analysis of (1) We construct a rst integral for (1) driven by the optimal strategy. Let 8 e D x ? s + m(eD + o) ln x ? a ln y + y for x > x1 ; > < D eD + o s V (x; y) = > eD + o (15) s 1 : : ?m ln x ? a ln y + y ? (ln ? 1) for x x D eD + o eD We note that the above function is continuous. Moreover, it is easy to see that the derivative of V along trajectories of (1) is zero almost everywhere and consequently, solutions of (1) which are driven by the optimal strategy are closed curves centered at E 2 . APPENDIX C: Qualitative analysis of (4) and (9) Characteristic polynomial corresponding to E 6 is
as + ma + as : 2 + mo o
Thus, due to the Routh-Hurwitz criterion, E 6 is locally asymptotically stable for (6) since we assume that all parameters are positive. We will prove that E 6 is globally stable using the Poincare-Bendixon theorem (Hartman 1964, Hofbauer and Sigmund 1984). To this end we have to exclude the existence of periodic orbits. This can be achieved by using the Dulac criterion (Hofbauer and Sigmund 1984) which is usually stated only for smooth vector elds (Hofbauer and Sigmund 1984), but it can easily be generalized for piecewise smooth vector elds (see also Busenberg and van den Driessche (1993)), which is the case of the system (4). Let us assume that eN > o. In the region x1N < x < x2N we have @ 1 ax ? xeN ? s y + @ 1 xeN ? s y ? my = ? s < 0 (16)
@x xy
o
@y xy
o
ox2
13 and in the region x2N < x
@ 1 (ax ? xy) + @ 1 (xy ? my) = 0: @x xy @y xy Assume there exists a periodic orbit of (4). If m < s=(eN ? o), equilibria E 5 , E 6 are to the left of the line x = x2N and every periodic orbit must partly belong to the region x1N < x < x2N .
However, due to (16), the Dulac criterion excludes existence of a periodic orbit and the Poincare{ Bendixon theorem implies that E 6 is globally asymptotically stable. If E 6 is to the right of the line x = x2N (which happens if m > s=(eN ? o)) we construct a Lyapunov function for (4). Let 8 for x > x2N < x? m ln x ? a ln y + y s V (x; y) = : 1 e x ? (mo + s) ln x + oy ? ao ln y + s ln 1 2 N o (eN ? o) ? s for xN x xN : The above function is continuous. For x > x2N it is the rst integral for the Lotka{Volterra equation (5). For x1N < x < x2N we have (f 6 denotes the right handside of (6)) mo + s 2 (e ? o) ye N N 0 6 2 hV ; f i = (x ? xN ) e ? x : o2 x N
Assuming m > s=(eN ? o) we note that (mo + s)=(eN ) > x2N . Thus for x1N < x < x2N the derivative of V along trajectories of (4) is negative for eN > o. Trajectories of (4) do converge to a global attractor bounded by the largest Lotka{Volterra cycle of (5) which is to the right of the line x = x2N . If eN < o then the host-parasitoid dynamics is described by (6) for x > x1N , and by (7) for x x1N . We see that the part of the (x; y) space for which x > x1N is invariant with respect to (4) and (16) holds. The Dulac criterion and the Poincare{Bendixon theory imply that E 6 is globally asymptotically stable. Now we consider non-concurrent and non-destructive host feeding. Characteristic polynomial corresponding to E 10 is 2 + m(e as+ o) + a(eN me ++eNoo + s) : N N Thus, due to the Routh-Hurwitz criterion, E 10 is locally stable provided we assume that all parameters are positive. To prove that E 10 is globally stable we use again the Dulac criterion to exclude closed orbits. We have s @ 1 ax ? xeN ? s y + @ 1 xeN ? s y ? my = ? @x xy eN + o @y xy eN + o (eN + o)x2 < 0: Thus there are no periodic orbits and E 10 is globally asymptotically stable. APPENDIX D: Behavior of the host-parasitoid model along the line x = x By n = (1; 0) we denote the normal vector to the line x = x . Denoting the right hand-side of (2) by f 2 and similarly for (6) we get (h; i stands for the scalar product)
hf 2; ni < 0 if y > a hf 6; ni > 0 if y < (e ae?D e ) : D N
14 It follows that trajectories of (12) driven by the optimal strategy cross the line x = x in direction from the left to the right if y < y1 and from the right to the left if y > y2 . In the segment with end points y1 and y2 trajectories are not uniquely determined by the initial condition, and they may move either to the left, or to the right. APPENDIX E: Qualitative analysis of (12) First, we assume eN eD =(eD ? eN ) > eN > o and we de ne 8 x ? m ln x ? a ln y + y for x > x2N > > > > s 1 > > < o eN x ? (mo + s) ln x + oy ? ao ln y + s ln (e ? o) ? s for x x x2N N V (x; y) = > e s D > > for x1D < x < x ? m + e + o ln x + e + o x ? a ln y + y + C1 > > > D D : ?m ln x ? a ln y + y + C for x < x1D 2 where constants C1 ; C2 are chosen in such a way that V is a continuous function. Function V is the rst integral for (2), (3) and (5). Assume eN ) m < e e s(?eDo? (17) (eD ? eN ) D N which implies that E 6 is to the left of the line x = x , i.e., mo + s < x :
eN
We get that for x < x < x2N ,
2 hV 0; f 6i = yeN o(2exN ? o) (x ? x2N ) moe + s ? x > 0: N
Therefore V is increasing along trajectories of (12) provided they are to the right of the line x = x which leads to destabilization of the dynamics. Now we consider the case m > s=(eN ? o). In this case for x < x < x2N we have hV 0 ; f 6 i < 0 since mo + s > x2 > x N eN and all trajectories of (12) do converge to a global attractor given by the largest Lotka{Volterra cycle of (12) which is to the right of the line x = x2N . Second, we assume eN eD =(eD ? eN ) > o > eN and we de ne 8 1 > for x x > > < o (eN x ? (mo + s) ln x + oy ? ao ln y) V1(x; y) = > ? m + s ln x + eD x ? a ln y + y + C3 for x1D < x < x > o eD + o > : ?m ln x ?eDa + ln y + y + C4 for x < x1D where constants C3 ; C4 are chosen in such a way that V is a continuous function. The stability analysis can be carried along the same lines as for the case eN > o, replacing function V by V1 . If (17) holds, we get that for x < x, mo + s 2 (e ? o) ye N N 2 0 6 (x ? x ) ?x >0 hV ; f i = 1
o2 x
N
eN
since x2N < 0, which leads to destabilization of those trajectories of (12) which enter the region x > x .
15
References Abrams, P. A. 1983. Foraging time optimization and interactions in food webs. Am. Nat. 124, 80{96. Aubin, J.-P. 1991. Viability theory. Birkhauser, Boston. Aubin, J. P. and A. Cellina 1984. Di erential inclusions. Grundlehren der mathematischen Wissenschaften 264, Springer-Verlag, Berlin. Briggs, C. J., R. M. Nisbet, W. W. Murdoch, T. R. Collier and J. A. J. Metz 1995. Dynamical e ects of host-feeding in parasitoids. Journal of Animal Ecology 64, 403{416. Busenberg, S. and P. van den Driessche 1993. A method for proving the non-existence of limit cycles. Journal of Mathematical Analysis and Applications 172, 463{479. Collier, T. R., W. W. Murdoch and R. M. Nisbet 1994. Egg load and the decision to host-feed in the parasitoid, Aphytis melinus. Journal of Animal Ecology 63, 299{306. Deimling, K. 1992. Di erential inclusions on closed sets. DeGruyter, Basel. Filippov, A. F. 1988. Di erential equations with discontinuous righthand sides. Kluwer Academic Publishers, Dordrecht. Flanders, S. E. 1953. Predatism by the adult hymenopterous parasite and its role in biological control. J. Econ. Ent. 46, 541{544. Hartman, P. 1964. Ordinary Di erential Equations. John Wiley& Sons, New York. Heimpel, G. E. and J. A. Rosenheim 1995. Dynamic host feeding by the parasitoid Aphitis melinus: the balance between current and future reproduction. Journal of Animal Ecology 64, 153{167. Hofbauer, J. and K. Sigmund 1984. The theory of evolution and dynamical systems. Cambridge University Press, Cambridge. Houston, A. I., J. M. McNamara and H. C. J. Godfray 1992. The e ect of variability on host feeding and reproductive success in parasitoids. Bulletin of Mathematical Biology 54, 465{ 476. Jervis, M. A. and N. A. C. Kidd 1986. Host-feeding strategies in Hymenopteran parasitoids. Biological Reviews 61, 395{434. Jervis, M. A. and N. A. C. Kidd 1995. Incorporating physiological realism into models of parasitoid feeding behaviour. Trends in Ecology and Evolution 10, 434{436. Kidd, N. A. C. and M. A. Jervis 1989. The e ects of host-feeding behaviour on the dynamics of parasitoid-host interactions, and the implications for biological control. Researches in Population Ecology 31, 235{274. Kidd, N. A. C. and M. A. Jervis 1991a. Host-feeding and oviposition by parasitoids in relation to host stage: Consequences for parasitioid-host population dynamics. Researches in Population Ecology 33, 87{99. Kidd, N. A. C. and M. A. Jervis 1991b. Host-feeding and oviposition strategies of parasitoids in relation to host stage. Researches in Population Ecology 33, 13{28.
16 Krivan, V. 1996. Optimal foraging and predator-prey dynamics. Theoretical Population Biology 49, 265{290. Krivan, V. 1997. Dynamic ideal free distribution: e ects of optimal patch choice on predatorprey dynamics. American Naturalist 149, 164{178. Krivan, V. and E. Sirot 1997. Searching for food or hosts: the in uence of parasitoids behavior on parasitoid-host dynamics. Theoretical Population Biology xx, xxx{xxx. Murdoch, W. W. and A. Stewart-Oaten 1989. Aggregation by parasitoids and predators: e ects on equilibrium and stability. American Naturalist 134, 288{310. Murdoch, W. W., R. M. Nisbet, R. F. Luck, H. C. J. Godfray and W. S. C. Gurney 1992. Sizeselective sex-allocation and host feeding in a parasitoid{host model. Journal of Animal Ecology 67, 533{541. Rosenheim, J. A. and D. Rosen 1992. In uence of egg load and host size on host-feeding behaviour of the parasitoid Aphytis lingnanensis. Ecological Entomology 17, 263{272. Sirot, E. and V. Krivan 1997. Adaptive superparasitsm and host-parasitoid dynamics. Bulletin of Mathematical Biology 59, 23{41. Stearns, S. T. 1992. The evolution of life histories. Oxford University Press, Oxford. Stephens, D. W. and J. R. Krebs 1986. Foraging theory. Princeton University Press, Princeton. Yamamura, N. and E. Yano 1988. A simple model of host-parasitoid interaction with hostfeeding. Researches in Population Ecology 30, 353{369.
17 FIGURES LEGENDS
Fig 1. Maximal tness for various host-feeding patterns is plotted as a function of host abundance. In Fig. 1A the parameters are such that o < eN eD =(eD ? eN ) which leads to the existence
of the critical host abundance x such that below it non-concurrent and destructive host-feeding gives the highest tness while above it the concurrent and non-destructive host feeding is the best choice. For o > eN eD =(eD ? eN ) see Fig. 1B. In this case the non-concurrent and destructive host-feeding pattern gives the highest parasitoid tness. Fig 2. For non-concurrent and destructive host feeding equilibrium E 2 is surrounded by closed curves. In this case host-feeding does not qualitatively in uence the neutral stability of the underlying Lotka{Volterra model. Parameters are as follows: a = 1:5; = 1; eD = 2; o = 1; m = 0:5; s = 1: Fig 3. For concurrent and non-destructive host feeding equilibrium E 6 is globally asymptotically stable if it is to the left of the line x = x2N (see Fig. 3A). In this simulation eN = 2, other parameters are the same as those given in Fig. 2. If E 6 is to the right of the line x = x2N (see Fig. 3B) then trajectories converge from outside to the largest Lotka{Volterra cycle of (5) which is to the right of the line x = x2N . In this simulation m = 1:5 and other parameters are the same as those given in Fig. 2. Fig 4. This gure shows the possible destabilization of the host-parasitoid dynamics when host-feeding pattern is chosen in an adaptive way. In Fig. 4A, equilibrium E 2 which is to the left of the line x = x is neutrally stable (since it is surrounded by a family of closed curves), but trajectories which move to the region where x > x do spiral away from the equilibrium. Parameters are as follows: a = 1:5; = 1; eD = 6; eN = 2; o = 1:3; m = 0:3; s = 1:5 In Fig. 4B, equilibrium E 6 which is to the right of the line x = x is locally, but not globally asymptotically stable. In this plot m = 1 and other parameters are the same as those given in Fig. 4A.
A
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Figure 1:
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Figure 2:
4
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Figure 3:
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Figure 4:
2.4
2.4
